Polar Coding
Polar codes, proposed by Arikan (in E. Arikan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels,” IEEE Transactions on Information Theory, vol. 55, pp. 3051-3073, July 2009) are the first class of constructive coding schemes that are provable to achieve the symmetric capacity of the binary-input discrete memoryless channels under a low-complexity successive cancellation (SC) decoder. However, the finite-length performance of polar codes under SC is not competitive compared to other modern channel coding schemes such as low-density parity-check (LDPC) codes and Turbo codes. Later, SC list (SCL) decoder is proposed by Tal and Vardy (in I. Tal and A. Vardy, “List Decoding of polar codes,” in Proceedings of IEEE Symp. Inf. Theory, pp. 1-5, 2011), which can approach the performance of optimal maximum-likelihood (ML) decoder. By concatenating a simple CRC coding, it was shown that the performance of concatenated polar code is competitive with that of well-optimized LDPC and Turbo codes. As a result, polar codes are being considered as a candidate for future 5G wireless communication systems.
The main idea of polar coding is to transform a pair of identical binary-input channels into two distinct channels of different qualities, one better and one worse than the original binary-input channel By repeating such a pair-wise polarizing operation on a set of 2M independent uses of a binary-input channel, a set of 2M “bit-channels” of varying qualities can be obtained. Some of these bit channels are nearly perfect (i.e. error free) while the rest of them are nearly useless (i.e. totally noisy). The point is to use the nearly perfect channel to transmit data to the receiver while setting the input to the useless channels to have fixed or frozen values (e.g. 0) known to the receiver. For this reason, those input bits to the nearly useless and the nearly perfect channel are commonly referred to as frozen bits and non frozen (or information) bits, respectively. Only the non-frozen bits are used to carry data in a polar code. Loading the data into the proper information bit locations have directly impact on the performance of a polar code. An illustration of the structure of a length-8 polar code is illustrated in FIG. 1 (example of polar code structure with N=8).
FIG. 2 illustrates an example of polar encoding with N=8. FIG. 2 shows the labeling of the intermediate info bits sl,i, where l∈{0, 1, . . . , n} and i∈{0, 1, . . . , N−1} during polar encoding with N=8. The intermediate info bits are related by the following equation:
                    s                              l            +            1                    ,          i                    =                        s                      l            ,            i                          ⊕                  s                      l            ,                          i              +                              2                l                                                          ,                  for        ⁢                                  ⁢        i            ∈                        {                                                    j                ∈                                  {                                      0                    ,                    1                    ,                    …                    ⁢                                                                                  ,                                          N                      -                      1                                                        }                                            :                              mod                ⁡                                  (                                                            ⌊                                              j                                                  2                          l                                                                    ⌋                                        ,                    2                                    )                                                      =            0                    }                ⁢                                  ⁢        and        ⁢                                  ⁢        l            ∈              {                  0          ,          1          ,          …          ⁢                                          ,                      n            -            1                          }                                s                              l            +            1                    ,                      i            +                          2              l                                          =              s                  l          ,                      i            +                          2              l                                            ,                  for        ⁢                                  ⁢        i            ∈                        {                                                    j                ∈                                  {                                      0                    ,                    1                    ,                    …                    ⁢                                                                                  ,                                          N                      -                      1                                                        }                                            :                              mod                ⁡                                  (                                                            ⌊                                              j                                                  2                          l                                                                    ⌋                                        ,                    2                                    )                                                      =            0                    }                ⁢                                  ⁢        and        ⁢                                  ⁢        l            ∈              {                  0          ,          1          ,          …          ⁢                                          ,                      n            -            1                          }            with s0,i=ut be the info bits, and sn,i≡xi be the code bits, for i∈{0, 1, . . . , N−1}.
For Polar code with distributed CRC, the input to the Polar encoder is first interleaved associated with the CRC polynomial. The information bits are interleaved, and a subset of CRC bits are distributed among the information bits.
The bit sequence c0, c1, c2, c3, . . . , cK-1 is interleaved into bit sequence c′0, c′1, c′2, c′3, . . . , c′K-1 as follows:
c′k=cπ(k), k=0, 1, . . . , k−1
where the interleaving pattern π(k) is given by the following:
if IIL = 0  Π(k) = k , k = 0,1,..., K − 1else k = 0;    for m = 0 to KILmax − 1      if ΠILmax (m) ≥ KILmax − K         Π(k) = ΠILmax (m) − (KILmax − K);         k = k + 1 ;      end if    end forend ifwhere πILmax(m) is given by TS 38.212, Table 5.3.1-1 (Interleaving pattern πILmax(m)), an example of which is shown in FIG. 3.New Radio Physical Broadcast Channel (NR-PBCH)
The 5G New Radio (NR) communication systems can operate with carrier frequencies ranging from hundreds of MHz to hundreds of GHz. When operating in very high frequency band, such as the millimeter-wave (mmW) bands (˜30-300 GHz), radio signals attenuate much more quickly with distance than those in lower frequency band (e.g. 1-3 GHz). Hence, in order to broadcast system information to user equipment (UE) over the same intended coverage area, beamforming is typically used to achieve power gain to compensate the path loss in high frequencies. Since the signal coverage of each beam can be quite narrow when many antennas are used to form the beam, the system information needs to be broadcast or transmitted at a different beam direction one at a time. This process of transmitting signal carrying the same information using beams with different (azimuth and/or elevation) directions one at a time is commonly referred to as beam sweeping. Since typically only one of the many beams carrying the same system information can reach a particular receiver with good signal strength, the receiver does not know the location of the received beam in the overall radio frame structure. In order to allow the receiver to determine the start and the end of a periodic radio frame, a time index is often included when broadcasting the system information through beam sweeping.
For example, FIG. 4 shows an example of how system information can be broadcast together with reference synchronization signal (SS) through beam sweeping. In this figure, the system information is carried by a physical channel called NR-PBCH which is transmitted in multiple synchronization blocks (SSB), each beamformed in a different direction. The SSBs are repeated within a certain NR-PBCH transmission time period (TTI, 80 ms in this example). Within a NR-PBCH TTI, the system information carried by NR-PBCH, MIB, in each SSB is the same. However, each NR-PBCH also carries a time index in order for the receiver to determine the radio frame boundaries. NR-PBCH may be encoded using Polar codes.
A preferred construction of the content of PBCH is shown below.
Number Informationof bitsCommentSFN10RMSI [8]Includes all information needed configurationto receive the PDCCH and PDSCH for RMSI including RMSI presence flag, RMSI/MSG2/4 SCS, possible QCL indication, and indication of initial active bandwidth part(if needed). 8 bits is the target with exact number of bits to be confirmed.SS block time index3Only present for above 6 GHzHalf frame indicationInclusion depends on agreement“CellBarred” flag11st PDSCH DMRS 1Working assumptionpositionPRB grid offset4Reserved bits[13] There will be at least 4 reserved (sub 6 GHz)bits. [10] In addition, reserved bits are (above 6 GHz)added to achieve byte alignment. Any additional agreed fields will reduce the number of reserved bitsCRC[19]Discuss if PBCH CRC should be PBCH-specific or aligned with PDCCH.Total including 56CRC